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Search: id:A008299
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| A008299 |
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Triangle of associated Stirling numbers of second kind. |
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+0
16
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| 1, 1, 1, 3, 1, 10, 1, 25, 15, 1, 56, 105, 1, 119, 490, 105, 1, 246, 1918, 1260, 1, 501, 6825, 9450, 945, 1, 1012, 22935, 56980, 17325, 1, 2035, 74316, 302995, 190575, 10395, 1, 4082, 235092, 1487200, 1636635, 270270, 1, 8177, 731731, 6914908, 12122110
(list; graph; listen)
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OFFSET
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2,4
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COMMENT
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Rows are of lengths 1,1,2,2,3,3,...
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
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LINKS
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L. M. Smiley, Completion of a Rational Function Sequence of Carlitz
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FORMULA
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S_r(n+1, k)=k S_r(n, k)+binomial(n, r-1)S_r(n-r+1, k-1) for this sequence, r=2 G.f.: sum(S_r(n, k)u^k ((t^n)/(n!)), n=0..infty, k=0..infty)=exp(u(e^t-sum(t^i/i!, i=0..r-1)))
a(n, k) = sum_{i=0..k} (-1)^i*binomial(n, i)*[sum_{j=0..k-i} (-1)^j*(k -i -j)^(n-i)/(j!*(k-i-j)!)] - David Wasserman (dwasserm(AT)earthlink.net), Jun 13 2007
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EXAMPLE
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There are 3 ways of partitioning a set N of cardinality 4 into 2 blocks each of cardinality at least 2, so S_2(4,2)=3.
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CROSSREFS
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Rows give A000247, A000478, A058844. Cf. A059022, A059023, A059024, A059025.
Row sums: A000296.
Sequence in context: A010289 A127613 A019427 this_sequence A016478 A102430 A135573
Adjacent sequences: A008296 A008297 A008298 this_sequence A008300 A008301 A008302
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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njas
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EXTENSIONS
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Formula and cross-references from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
More terms from David Wasserman (dwasserm(AT)earthlink.net), Jun 13 2007
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